Solve for $k$, $ \dfrac{1}{15k - 15} = -\dfrac{1}{6k - 6} - \dfrac{4k + 4}{3k - 3} $
Solution: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15k - 15$ $6k - 6$ and $3k - 3$ The common denominator is $30k - 30$ To get $30k - 30$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{1}{15k - 15} \times \dfrac{2}{2} = \dfrac{2}{30k - 30} $ To get $30k - 30$ in the denominator of the second term, multiply it by $\frac{5}{5}$ $ -\dfrac{1}{6k - 6} \times \dfrac{5}{5} = -\dfrac{5}{30k - 30} $ To get $30k - 30$ in the denominator of the third term, multiply it by $\frac{10}{10}$ $ -\dfrac{4k + 4}{3k - 3} \times \dfrac{10}{10} = -\dfrac{40k + 40}{30k - 30} $ This give us: $ \dfrac{2}{30k - 30} = -\dfrac{5}{30k - 30} - \dfrac{40k + 40}{30k - 30} $ If we multiply both sides of the equation by $30k - 30$ , we get: $ 2 = -5 - 40k - 40$ $ 2 = -40k - 45$ $ 47 = -40k $ $ k = -\dfrac{47}{40}$